Problem
b) Solve the linear system below using the \( L U \) decomposition method. \[ \begin{aligned} x_{1}+x_{2}+3 x_{4} & =8 \\ 2 x_{1}+x_{2}-x_{3}+x_{4} & =7 \\ 3 x_{1}-x_{2}-x_{3}+2 x_{4} & =14 \\ -x_{1}+2 x_{2}+3 x_{3}-x_{4} & =-7 \end{aligned} \]
Solution
Step 1 ：Perform LU Decomposition of the coefficient matrix A: \( A = L U \)
Step 2 ：Solve for y using forward substitution: \( L y = b\)
Step 3 ：Solve for x using backward substitution: \( U x = y\)
